The diagram depicts the curve $y = (\ln x)^2$. The $x$-coordinate of $P$ is $e$, and the normal to the curve at $P$ intersects the $x$-axis at $Q$.
(i)[4]
Determine the $x$-coordinate of $Q$.
(ii)[1]
Show that $\int \ln x\,dx = x\ln x - x + c$, where $c$ is a constant.
(iii)[5]
Using integration by parts, or by another method, determine the exact area of the shaded region bounded by the curve, the $x$-axis and the normal $PQ$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply derivative is $\dfrac{2\ln x}{x}$” …